# Leeson's equation

Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum.

Leeson's expression[1] for single-sideband (SSB) phase noise in dBc/Hz (decibels relative to output level per Hertz) and augmented for flicker noise:[2]

${\displaystyle L(f_{m})=10\log {\bigg [}{\frac {1}{2}}{\bigg (}{\bigg (}{\frac {f_{0}}{2Q_{l}f_{m}}}{\bigg )}^{2}+1{\bigg )}{\bigg (}{\frac {f_{c}}{f_{m}}}+1{\bigg )}{\bigg (}{\frac {FkT}{P_{s}}}{\bigg )}{\bigg ]}}$

where f0 is the output frequency, Ql is the loaded Q, fm is the offset from the output frequency (Hz), fc is the 1/f corner frequency, F is the noise factor of the amplifier, k is Boltzmann's constant in Joules/Kevin, T is absolute temperature in Kelvins, and Ps is the output power.[3]

## References

1. ^ Leeson, D. B. (February 1966), "A Simple Model of Feedback Oscillator Noise Spectrum", Proceedings of the IEEE, 54 (2): 329–330, doi:10.1109/PROC.1966.4682
2. ^ Rhea, Randall W. (1997), Oscillator Design & Computer Simulation (Second ed.), McGraw-Hill, ISBN 0-07-052415-7, p. 115.
3. ^ Zumbahlen, Hank; Analog Devices, Inc. (2011-08-30). Zumbahlen, Hank, ed. Linear Circuit Design Handbook. Newnes. ISBN 9780080559155.